A barrelledness criterion for C0(E)

Abstract

Denote by C0(E) the linear space of all sequences in E which converge to zero in E endowed with its natural topology. A. Marquina and J. M. Sanz Serna [same journal 31 (1978/79), 589–596; MR0531574 (80i:46010)] gave necessary and sufficient conditions to ensure the quasibarrelledness of C0(E), namely: (i) E is quasibarrelled and (ii) the strong dual of E satisfies condition (B) of Pietsch. Moreover, if E is complete in the sense of Mackey, (i) and (ii) characterize the barrelledness of C0(E). Even in the absence of completeness in the sense of Mackey, the last result is true as the article under review shows by a clever use of a "sliding-hump'' technique. Interesting consequences of those results in the theory of tensor products and spaces of vector-valued continuous functions can be found in a recent paper by A. Defant and W. Govaerts ["Tensor products and spaces of vector-valued continuous functions'', Manuscripta Math., to appear] and in the monograph of J. Schmets [Spaces of vector-valued continuous functions, Lecture Notes in Math., 1003, Springer, Berlin, 1983].